Experimental Determination of Average and Equilibrium Structures of Polyatomic Gas Molecules by Diffraction and Spectroscopic Methods BY KOZOKUCHITSU AND KAZUKO OYANAGI Department of Chemistry, Faculty of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113, Japan Received 29th April, 1976 Various definitions and interrelations of the " geometric structures " of polyatomic molecules are reviewed. Analytical procedures for determining well-defined " structures " are discussed. The variations in average nuclear positions caused by isotopic substitution and their influence on the structure are estimated for OCS, HCN, SO2 and H20 as examples. 1. GENUINE AND APPROXIMATE EQUILIBRIUM STRUCTURES FROM SPECTROSCOPY The geometric structure of a semirigid polyatomic molecule can best be defined and compared with the corresponding theoretical structure in terms of the nuclear positions at a potential minimum.A set of these " equilibrium " nuclear coordinates, or the internuclear distances and angles derived therefrom, are called the re-structure. When the rotational constants for the ground vibrational state and those for the excited states of all the fundamental modes are determined, they can be extrapolated to give the equilibrium rotational constants, A,, Be and Ce, from which the re structure can be determined. This method has been applied to a number of linear and bent XY, molecules.1$2 For many other molecules, where the number of independent geometric parameters exceeds that of independent rotational constants, a sufficient number of isotopic rotational constants is also necessary.The re-structures of several linear XYZ, pyramidal XY3 and CH3X molecules have been determined in this way.2 In principle, a similar procedure may be applied to a nonrigid molecule in order to define the equilibrium structure of a semirigid frame as a function of the large-amplitude coordinates. However, no complete experimental analysis of this sort seems to have been made. In most other cases, only the ground-state rotational constants, A,, Bo and C,, are used to derive geometric parameters. Because of complicated vibration-rotation interactions3 however, these rotational constants do not correspond exactly to the ground-state average of nuclear positions. Hence, the so-called r,-structure derived from A,, Bo and Co is not necessarily a good approximation of the re-structure.The experimental precision of ro parameters is often significantly lower than that of the rotational constants. This systematic error can be even larger when some of the geometric parameters have to be assumed. When one of the component nuclei is substituted by its isotope, the isotopic differences between the rotational constants of this species and the parent species can be used to determine the principal-axis substitution (r,) coordinates of this nucleus, whereby a substantial part of the vibration-rotation interaction is compensated.KOZO KUCHITSU A N D KAZUKO OYANAGI 21 Kraitchman’s equations are used for this purpo~e.~*~ When the rotational constants of all the singly substituted species are determined, a complete r,-structure is obtained.Even when complete isotopic substitutions are not practicable, one can use a first- moment equation, or the condition that the cross-products of inertia vanish, to esti- mate the remaining nuclear coordinates. In a less favourable case the rotational constants of the parent species have to be used to obtain a hybrid r,,r,-structure at the expense of a systematic error caused by vibration-rotation interaction. The precision of the Y, coordinates depends on the nuclear positions. The co- ordinates of a nucleus located close to a principal inertial plane carry large uncertain- ties irrespective of the nuclear mass; an empirical equation for this estimation has been proposed by Costains In addition, the signs of such coordinates may be ambiguous, since Kraitchman’s equations provide only their absolute values.Doubly substituted species have sometimes been used to improve the precision of such small coordinates .6 The difference between the r, and re-structures of polyatomic molecules has not been understood as clearly as that for diatomic molecules, though physical significance of the r, coordinates has been discussed by Sorensen et aL7 and Watson.* There still are only a limited number of molecules for which both the r, and re-structures have been determined precisely from experiment and compared with each other ~ritically.~.~ The rs parameters, such as the C-N bond lengths in various cyanides, determined with an empirical correction for vibration-rotation interaction appear to be nearly independent of the isotopic species used to determine the coordinates, and are chemically pla~sible.~ Nevertheless, systematic errors in the Y, parameters should be treated cautiously when one intends to make a critical comparison of r, parameters in different geometric environments for chemical purposes.For example, recent electron diffraction studies suggest that the observed differences in the r, or ro bond lengths in some simple organic molecules are not necessarily close approximations of those in the corresponding re bond lengths.l0-l2 When the Y, coordinates of all the atoms are determined, one can calculate a moment of inertia, I,, using these rs coordinates. Watson8 has pointed out that the equilibrium moment of inertia, I,, is approximately equal to 21,-10, where I, is h/8n2 divided by the corresponding rotational constant for the ground vibrational state.He called the structure derived from 2Zs-10 the mass-dependence (rm) structure. In a few examples he has shown that the r,-structure is a very close approximation of the re-structure except for some parameters involving hydrogen. Since, however, one needs rotational constants for more isotopic species than are necessary for determining the r,-structure, the application of his method has so far been restricted to very simple molecules. 2. ZERO-POINT AVERAGE STRUCTURE FROM SPECTROSCOPY The difference between the ground-state and equilibrium rotational constants is composed of the harmonic part, which depends on the quadratic potential constants, and the anharmonic part, which is a linear function of the cubic constant~.~J~ When the harmonic part is subtracted from the ground-state rotational constants, one obtains the zero-point average rotational constants, A,, B, and C,, which correspond to the zero-point average (r,) nuclear positions.14-16 The r,(X-Y) distance between the nuclear positions X and Y in a polyatomic molecule should not be confused with the zero-point vibrational average of an instantaneous X - Y distance.A simple calculation 1 2 9 1 7 shows that for a molecule with small vibrational amplitudes rz = r e + < W * , (2.1122 EXPERIMENTAL DETERMINATION OF STRUCTURES OF GAS MOLECULES where Az denotes an instantaneous displacement, Ar, of r(X-Y) projected on the equilibrium X-Y axis (taken as a temporary z axis), and 0 denotes an average over the ground vibrational state.Because nuclear vibrations perpendicular to this z-axis exist in a polyatomic molecule, r, does not agree with the real average X-Y distance, re+<Ar),, which is denoted as r,O. The difference between r, and re, i.e., (Az),, can be estimated if cubic anharmoni- city18 is known or assumed. On the other hand, the difference between (Ar), and (Az), can be estimated with sufficient accuracy if the quadratic force field is known (see section 3). The average bond angle can be defined unambiguously in terms of the average nuclear positions. The r, position depends on nuclear masses. Therefore, the isotopic effect must be known precisely when the A,, Bz and C, of other isotopic species are needed for a complete determination of the r,-structure. The r,-structure derived from such isotopic substitutions is sensitive to the estimated (or assumed) isotopic effect on r,- positions.This presents a serious problem in the experimental determination of the r,-stru~ture.~~ In most previous studies the isotopic variation of an r, parameter, 6r, NN ~(Az),, was confused with Thus the dr, of a bond by substitution of one of the atoms by its heavier isotope was erroneously called " bond shrinkage " or " bond shortening ". This confusion is by 110 means trivial, as discussed more quantitatively in section 4. 3. AVERAGE STRUCTURES FROM ELECTRON DIFFRACTION Gas electron diffraction provides information on the thermal average of all the internuclear distances in a molecule.The rg distance is defined aslo where T denotes the vibrational average in thermal equilibrium. It has been shown that17 where KT = ((Ax2> T + (Ay2> T)/2re (3.3) Ax and Ay denote the displacements perpendicular to the equilibrium internuclear axis z, drcent denotes a small displacement due to centrifugal force. One advantage of rg over rz is that the re bond distance can be estimated if bond-stretching anharmoni- city is assumed.21*22 For example, for a group of similar bonds (e.g., for the C-C bonds in hydrocarbons), thermal average displacements, <Ar)=, can be estimated to be nearly equal to one another, so that the observed differences in the rg distances may well be approximated as those in the re distances. On the other hand, since the KT terms do not satisfy geometric conditions, a set of the rg bonded and nonbonded distances cannot define a physically meaningful bond angle without corrections for KT, so-called linear or nonlinear " shrinkage effects ".23 For example, the rg (Y-Y) distance in a linear XY2 molecule is smaller than twice the rg (X-Y) distance.However, such a shrinkage effect can be calculated if the quadratic force field is The thermal average of nuclear positions, ra, derived from the thermal average of internuclear distances, rg, is defined for this purpose;1°J7 rg = rg--KT-drCent. (3.4)KOZO KUCHITSU AND KAZUKO OYANAGI 23 The ra distance corresponds to the distance between thermal average nuclear positions, and when it is extrapolated to zero kelvin, rao = lim rCr = r,O-Ko z r,+(Az),, T-tO the rao structure is practically identical with the r, structure.This formalism provides a basis for an analysis of experimental data derived from electron diffraction in combination with rotational constants in order to determine a consistent average s t r ~ c t u r e . ~ ~ * ~ ~ For a relatively complicated molecule, where nearly equal and inequivalent internuclear distances exist, it is often difficult for electron diffraction alone to determine all the independent geometric parameters accurately without assumptions. In such a case, rotational constants supply precise independent information on geometric parameters. On the other hand, electron diffraction gives important information for the determination of a unique and accurate average structure, which may not be derived from spectroscopy alone even with numerous isotopic substitutions.So far, only the rotational constants of the parent species or those of deuterated species have been used for such a combined analysis, since for non-hydrogen substi- tutions the isotopic variations, Sr,, mentioned in section 2 are so uncertain that their rotational constants cannot improve the accuracy of the derived parameters. For example, in a structure analysis of a~rolein,~' H2C=CH-CH=0, only the rotational constants of the parent species and a part of the D species were used, whereas those of the 13C and l 8 0 species had to be left out. Under these circumstances, it is pertinent to make a reasonable estimate of Sr, in order to make full use of the rotational constants of all the available isotopic species.4. ISOTOPIC DISPLACEMENTS OF NUCLEAR POSITIONS Average nuclear positions in the principal inertial axes, (ai),,, (bi)o and ( c ~ ) ~ , For example, it is known (4.1) are linear functions of the cubic potential constants,13 kSSrSIt. that (ai>o = a4+(Aa,)o = a:+mi't Clci.,' <Qs>o S where a: is the equilibrium position of the nucleus i and Zc$ is an element of the I matrix. The ground-state vibrational average of the normal coordinate of the s'th mode, (Qs)o, is given by28 where us is the normal frequency and g,. is the degeneracy. When the nucleus j is substituted by its isotope, the displacements of the average position of the nucleus i, 6ai = (ai*>-(ai>, etc., (4.3) can be calculated if one assumes that the potential surface remains unchanged.The nucleus i may or may not be equal to j . It follows that (a,*> = a;+ml-* 2 Z(i",'* <Qs*), S (4.4) where the starred quantities can be calculated by use of the quadratic and cubic potential constants obtained with appropriate mass corrections. The isotopic variations in the r, distances and angles, 6r, and SO,, can then be calculated. Numerical examples of four triatomic molecules, for which precise rotational24 EXPERIMENTAL DETERMINATION OF STRUCTURES OF GAS MOLECULES constants, equilibrium structures, and quadratic and cubic potential constants are known (OCS,29 HCN?O S031 andH2032*33), are listed in fig. 1-4. The following remarks can be made on the isotopic variations: S -3.3 -1.3 ~ o* -11.0 c* 4.4 S* o* -126 :* -7.1 -9.5 c* -2.9 s* -7.0 :*-O.l x x -1.6 -28 s* FIG.1.-Isotopic differences in the I;. distances from those in 16012C32S, in units of A. The asterisks represent lSO, 13C and 34S. The arrows indicate that additivity rules hold in multiple isotopic substitutions. - N* H - ' O C - L $ - t HAC* -7 ,: ;14 :* -3 >c, 435 c* -108, N* N +309 -101 "x ;4 , +296 :*-lo3 FIG. 2.-Isotopic differences in the rz distances from those in H12C14N in units of lW5 A. The asterisks represent 13C and lSN. The arrows indicate that additivity rules hold in multiple isotopic substitutions. FIG. 3.-Isotopic differences in the rz distances (in lW5 A) and the BZ angle (in lW3 degrees) from those in 32S1602. The asterisks represent ''0 and "S. The arrows indicate that additivity rules hold in multiple substitutions.0 - 5 ? 4 . 6 k 7 \ -137&~22 O* D D D FIG. 4.-Isotopic differences in the rz distances (in lW5 A) and the 8, angle (in low3 degrees) from those in H2160. The asterisks represents "0. The arrows indicate that additivity rules hold approxi- mately in multiple substitutions. (i) The orders of magnitude of the isotopic variations of r, bond distances are a few thousandths of 1 A when H/D substitution is made, and a few ten thousandths of 1 A or less in other cases. (ii) The dr, caused by multiple isotopic substitutions are additive. For example, the variations of the r, distances in 18013C34S, from 16012C32S is approximately equal to the sum of those in 18012C32S, 16013C32S and 16012C34S from 16012C32S. (iii) When a nucleus is substituted by its heavier isotope, 6rz may not necessarilyKOZO KUCHITSU AND KAZUKO OYANAGI 25 be negative.A typical example is an apparent increase in the r, distance of C-D in DCN in comparison with that of C-H in HCN.* (iv) When a nucleus X is substituted, the 6r, of a bond which is not directly associated with X, say Y-Z, can be comparable with, or even larger than, that directly associated with X, say X-Y. A typical example of this " secondary isotopic varia- tion " is shown in the large dr, (C-S) in OCS when l6O is substituted by "0. The trends (iii) and (iv) are explicable by use of eqn (2.1) and (3.5), Since the first and the second terms have the same sign (positive or negative, respec- tively, when one of the nuclei is substituted by its lighter or heavier isotope), they tend to cancel each other.The secondary variation is mainly ascribable to 6Ko, which accounts for the isotopic difference in perpendicular displacements (see section 2). The second term can outweigh the first term and, in case of substitution with a heavier isotope, make dr, positive. In other words, an apparent " bond stretch " instead of a " bond shortening " can take place. 5. THE r z STRUCTURE DERIVED FROM ISOTOPIC SUBSTITUTION The rotational constants, A,, B, and C,, for various isotopic species can be used to determine the substitution r, coordinates. A schematic diagram is shown in fig. 5. The isotopic displacements in the nuclear coordinates, 6ai, 6bi and 6ci, are I I I 9 structure re Is h., 1 r.7 FIG. 5.-A schematic diagram showing determination of structures from isotopic rotational constants.introduced into the substitution scheme. The Kraitchman coordinates derived therefrom has been formulated by Nygaard and S ~ r e n s e n . ~ ~ The zero-point average coordinate, a,,, derived from the Kraitchman equation with inclusion of the isotopic displacements in all the hai caused by the substitution of a nucleusj is given as a linear combination of 6ai by \ i J where a, jeff denotes the corresponding effective coordinate calculated by a direct application of the Kraitchman equation4 to A,, B, and C, with neglect of 6ai, and m and Am denote the atomic mass and the isotopic difference. The sum is taken over all the atoms including j . The r,-structure can then be calculated from the a,, co- ordinates.The isotopic displacement is multiplied by a large factor particularly when a, is small and/or miaI is large. * The difference, rz(C-D)-r,(C-H), was estimated by Laurie and Herschbach16 to be -0.003A. The discrepancy between their estimate and the present estimate, +0.003A, seems to originate from their neglect of the l2C-I3C isotope differences in the v, parameters when they estimated the r,(C-H) and r,(C-D) distances independently by use of the B, constants of the 12C and "C species.26 EXPERIMENTAL DETERMINATION OF STRUCTURES OF GAS MOLECULES This method has been applied to OCS, HCN, SO2 and H20, as summarized in table 1. The r,-structures of SOz and H20 obtained by the present substitution method are consistent with those derived directly from the A, and B, (or C,) * of the parent molecules, and the r,-structures of OCS and HCN are consistent with any of those derived from the B, constants of any two isotopic species.For comparison, TABLE 1 .-STRUCTURES OF TRIATOMIC MOLECULES * ocs 0-c c-s HCN H-C C-N SO2 s-0 L 0-s-0 D20 t D-0 L D-0-D 1.1543 1.5628 1.0655 1.1532 1.4308 0.9575 119.33 104.51 0.0063 - 0.003 1 - 0.0023 0.0019 0.0015 0.07 0.0039 -0.51 0.003 1 0.001 7 0.0046 0.0038 0.10 0.0107 - 0.02 - 0.0043 0.0065 -0.0031 - 0.0028 0.0026 0.0022 0.10 0.0067 -0.25 * Distances in 8, and angles in degrees. The Y, and rS structures are taken from the references cited in the text. The Y, and rZeff-structures are calculated with and without the isotopic variations, 6r,, given in fig. 1-4, respectively. 7 The r,-structure of D20 has been calculated by use of the rotational constants of D2160, H2l60 and D2'*0.the r,eff-structures derived from azjeff coordinates are also listed in the table. They are close to the r,-structures derived from A. and Bo by use of the ordinary Kraitchman method with neglect of isotopic displacements. The differences between the r, and rZeff distances are a few thousandths of 1 A. Though none of the molecules has small nuclear coordinates (0.1 or less), the differences are one order of magnitude < dr, and are comparable with experimental precision. 6. EXTENSION TO LARGER MOLECULES The rigorous method stated above for estimation of isotopic variations, 6r,, requires precise knowledge of the quadratic and cubic potential constants. Even if the cubic constants are not available, one can use an approximate anharmonic model for this estimation.The simplest model seems to be to use the quadratic force field and the Morse anharmoiiicity parameters, a, of the bonds estimated from the cor- responding diatomic m01ecules.~~*~~ The dr, values for the four molecules listed in fig. 1-4 have been estimated by use of this model. The orders of magnitude of the dr, distances have been estimated correctly, but the agreement is not always quantitative. Furthermore, the isotopic variations in the angles have to be estimated somehow. The potential constants obtained from ab initio calculations, if available, may be used for this purpose. In this respect, the isotopic variations of the r,-structures of H2C0 and C2H4 calculated by Duncan35 from the observed isotopic rotational constants are suggested.The average structure determined by electron diffraction can also supply inde- pendent and accurate information on the dr, parameters. In such favourable cases as CH3CN36 and V0C13,37 dr, parameters can be determined as a part of the variable parameters in a least-squares analysis. The subject of this section will be reported in detail elsewhere. * In principle, there is no inertial defect among A,, B, and C, for a planar molecule.KOZO KUCHITSU AND R A Z U K O OYANAGJ 27 7. SUMMARY A spectroscopic experiment primarily determines nuclear positions, either equili- brium or vibrational average, whereas electron diffraction primarily determines thermal-average internuclear distances. These " position " and " distance " averages can be converted from one to the other if the quadratic force field of the molecule is known at least approximately. A conjoint analysis of electron diffraction data and rotational constants can improve the accuracy of the average structure.When rotational constants of different isotopic species are used to determine the average structure, it is necessary to consider the isotopic dependence of average nuclear positions. Our main future problems are the following: (i) Estimation of the equilibrium structure from average structures. For this purpose, our potential models have to be refined. Particularly, accurate data from high-resolution spectroscopy and from ab initio calculations should be most helpful.(ii) A complementary use of accurate electron diffraction data. There is much to be investigated in this respect before one understands spectroscopic structures of medium-sized molecules, either Y, or ro. (iii) Structure analyses of nonrigid molecules. In order to apply the present method to nonrigid molecules, the theory needs to be modified and the potential function in regard to the large-amplitude motion has to be known. A combined analysis of electron diffraction and microwave spectroscopic data for symmetric internal rotors has been formulated by Iijima.38 Efforts are being made to work out a more general formulation. The authors are grateful to Drs. Lise Nygaard and G. Ole Slarensen, Chemical Laboratory V, University of Copenhagen, for helpful discussions.' Y . Morino, Pure Appl. Chem., 1969,18, 323. Y . Morino and E. Hirota, Ann. Rev. Phys. Chem., 1970, 20, 139. I. M. Mills, Molecuiar Spectroscopy, Modern Research, ed. K. N. Rao and C. W. Mathews (Academic Press, New York, 1972), p. 115. J. Kraitchman, Amer. J. Phys., 1953, 21, 17. C. C. Costain, J. Chem. Phys., 1958, 29, 864; Trans. Amer. Cryst. Assoc., 1966, 2, 157. L. Pierce, J. Mol. Spectr., 1959, 3, 575. J. K. G . Watson, J, Mol. Spectr., 1973, 48, 479. R. H. 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